Quantile of a Mixture

TL;DR

This paper derives an explicit formula for the quantile of a mixture of two random variables, aiding in risk assessment when limited information is available.

Contribution

It provides a comprehensive analysis of the quantile of mixtures, including discrete and continuous cases with unbounded support, enhancing risk management tools.

Findings

Explicit formula for mixture quantiles derived

Applicable to discrete and continuous variables with unbounded support

Facilitates bounds on Value-at-Risk with partial information

Abstract

In this note, we give an explicit expression for the quantile of a mixture of two random variables. We carefully examine all possible cases of discrete and continuous variables with possibly unbounded support. The result is useful for finding bounds on the Value-at-Risk of risky portfolios when only partial information is available (Bernard and Vanduffel (2014)).